twisted filters

Nick Witheley (Bristol) and Anthony Lee (Warwick) just posted an interesting paper called ‘Twisted particle filters‘ on arXiv. (Presumably unintentionally, the title sounds like Twisted Sister, pictured above, even though I never listened to this [particularly] heavy kind of hard rock! Twisting is customarily used in the large deviation literature.)

The twisted particle paper studies the impact of the choice of something similar to, if subtly different from, an importance function on the approximation of the marginal density (or evidence) for HMMS. (In essence, the standard particle filter is modified for only one particle in the population.) The core of the paper is to compare those importance functions in a fixed N-large n setting. As in simpler importance sampling cases, there exists an optimal if impractical choice of importance function, leading to a zero variance estimator of the evidence. Nick and Anthony produce an upper bound on the general variance as well. One of the most appealing features of the paper is that the authors manage a convergence result in n rather than N. (Although the algorithms are obviously validated in the more standard large N sense.)

The paper is quite theoretical and dense (I was about to write heavy in connection with the above!), with half its length dedicated to proofs. It relies on operator theory, with eigen-functions behind the optimal filter, while not unrelated with Pierre Del Moral’s works. (It took me a while to realise that the notation ω was not the elemental draw from the σ-algebra but rather the realisation of the observed sequence! And I had to keep recalling that θ was the lag operator and not a model parameter [there is no model parameter].)